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\begin{document}

\title{Electrical Measurement of the Direct Spin Hall Effect in
Fe/In$_{x}$Ga$_{1-x}$As Heterostructures}



\author{E. S. Garlid}
\affiliation{School of Physics and Astronomy, University of Minnesota,
Minneapolis, MN 55455}
\author{Q. O. Hu}
\affiliation{Dept. of Electrical and Computer Engineering, University
of California, Santa Barbara, CA 93106}
\author{M. K. Chan}
\affiliation{School of Physics and Astronomy, University of Minnesota,
Minneapolis, MN 55455}
\author{C. J. Palmstr\o m}
\affiliation{Dept. of Electrical and Computer Engineering, University
of California, Santa Barbara, CA 93106}
\affiliation{Dept. of Materials, University of California, Santa
Barbara, CA 93106}
\author{P. A. Crowell}
\affiliation{School of Physics and Astronomy, University of Minnesota,
Minneapolis, MN 55455}
%\email[]{crowell@physics.umn.edu}


\begin{abstract}
We report on an all-electrical measurement of the spin Hall effect in
epitaxial Fe/In$_{x}$Ga$_{1-x}$As heterostructures with $n$-type
channel doping (Si) and highly doped Schottky tunnel barriers.   A
transverse spin current generated by an ordinary charge current
flowing in the In$_{x}$Ga$_{1-x}$As is detected by measuring the spin
accumulation at the edges of the channel. The spin accumulation is
identified through the observation of a Hanle effect in the Hall
voltage measured by pairs of ferromagnetic contacts. We investigate
the bias and temperature dependence of the resulting Hanle signal and
determine the skew and side-jump contributions to the total spin Hall
conductivity.
\end{abstract}
\pacs{72.25.Dc,72.25.Rb, 85.75.-d}
\maketitle

The generation and manipulation of spin populations in a single device
by making use of spin-orbit coupling has been a longstanding goal of
the field of semiconductor spintronics
\cite{D'yakonov:PLA:1971,Datta:APL:1990}.  There has been extensive
theoretical discussion of the spin Hall effect (SHE) and the various
ways that it could be exploited to generate or manipulate spin
currents
\cite{Sinova:PRL:2004,Schliemann:PRB:2004,Engel:PRL:2005,
Bernevig:PRB:2005,Tse:PRL:2006,Engel:PRL:2007}.  However, only a
handful of recent experiments have investigated this effect, and in
semiconductor materials they have relied on optical techniques to
either detect
\cite{Kato:Science:2004,Wunderlich:PRL:2005,Sih:NatPhys:2005,
Stern:PRL:2006,Matsuzaka:PRB:2009} or generate
\cite{Wunderlich:NatPhys:2009} spins.   The scope of experimental
studies could be broadened significantly by access to transport
techniques that can probe both materials and device geometries that
are not accessible optically.

In this Letter we report on an all-electrical measurement of the SHE
in lateral devices fabricated from Fe/In$_{x}$Ga$_{1-x}$As
heterostructures doped just above the metal-insulator transition.  The
SHE is due to spin-orbit scattering of an ordinary charge current,
resulting in a transverse spin current.  In the geometry shown in
Fig.~\ref{fig:fig1}(a), a charge current $j_{x} = \sigma_{xx}E_{x}$
flows down a channel of conductivity $\sigma_{xx}$  in the presence of
an electric field $E_{x}$.  The electrons have a drift momentum $\hbar
{\mathbf k} = (m^{*} j_{x}/ne) \hat x$, where $n$ is the carrier
density and $m^{*}$ is the effective mass.   The electron spins
interact with impurities via the spin-orbit Hamiltonian
%\begin{equation}
$H_{so} = \lambda_{so}{\boldsymbol \sigma}\cdot({\mathbf k}\times
\nabla V),$
%\label{eq:SO}
%\end{equation}
where $\lambda_{so}$ is the spin-orbit coupling constant, $\boldsymbol
\sigma$ is the Pauli spin operator, and  $V$ is the Coulomb-like
scattering potential.  The scattering process leads to spin-dependent
deflection of electrons, resulting in a spin current $j_{s}$
perpendicular to both their spin orientation and drift momentum.  In
steady state, this process leads to an accumulation of spins of
opposite sign at the two edges of the channel, and we detect this spin
accumulation using Fe contacts as spin-dependent Hall voltage probes. 
 For the geometry shown in Fig.~\ref{fig:fig1}(a), we are sensitive to
a spin current $j_{s}$ that flows in the $y$ direction, with the spin
oriented along $z$.   We find that the magnitude of the spin Hall
conductivity $\sigma_{SH}=j_{s}/E$ is in agreement with models of the
extrinsic SHE due to ionized impurity scattering
\cite{Engel:PRL:2005,Tse:PRL:2006}.  By analyzing the dependence of
the SH signal on channel conductivity, we determine the relative
magnitudes of the skew and side-jump contributions to the total spin
Hall conductivity.  We find that the ratio of these terms is
approximately constant, independent of the spin orbit coupling
parameter, but  the relative magnitude of the side-jump contribution
is consistently larger than predicted by theory
\cite{Engel:PRL:2005,Tse:PRL:2006}.  The temperature dependence of the
spin Hall conductivity is weak over the range of our experiment ($T <
150$~K), although our sensitivity at the highest temperatures is
limited by the short spin lifetime in the channel.

Epitaxial (001) Fe/In$_{x}$Ga$_{1-x}A$s heterostructures were grown by
molecular beam epitaxy.  The semiconductor epilayers were 2.5~$\mu$m
thick and with Si-dopings between $n = 3 \times 10^{16}$~to~$5 \times
10^{16}$~cm$^{-3}$.  Highly doped Schottky tunnel barriers ($n^{+} = 5
\times 10^{18}$~cm$^{-3}$) were prepared as described in
Ref.~\onlinecite{Lou:NatPhys:2007}.  Four heterostructures with In
concentrations $x =$~0.00, 0.03, 0.05, and 0.06 were studied.  The
wafers were subtractively processed into devices using standard
lithographic and etching techniques \cite{Lou:NatPhys:2007}.  Multiple
devices were fabricated on a single chip with 30$~\mu$m wide channels
oriented along the [110] direction, which is the $x$-direction in the
schematic diagram of Fig.~\ref{fig:fig1}(a).  Pairs of Fe electrodes,
each of which is 4~$\mu$m wide, were patterned so that the centers of
the contacts in each pair are 2, 6, or 10$~\mu$m from the edges of the
channel. The device geometry is shown schematically in
Fig.~\ref{fig:fig1}(a), in which each of the contact separations is
illustrated on a single device for simplicity.  Current injection
contacts are located at the ends of the channel, $> 250~\mu$m away
from the Hall contacts.  The charge current $j_x$ is therefore
unpolarized.
\begin{figure}
    \includegraphics*[width=8.5cm]{Figure1_Garlid.eps}
    \caption{(color on-line) (a) Schematic diagram (not to scale) of
the device layout and SHE measurement.  In the actual experiment,
contacts at different edge separations are on different devices on the
same chip and the current injection contacts are located $> 250~\mu$m
away from Hall measurement contacts.  (b) Non-local spin valve
($\relbar$) and Hanle effect ($\bullet$) data obtained on a GaAs
device at $T=60$~K for $j_{Inj} = 8.2\times 10^2$~A/cm$^2$. Hanle data
are shown for both parallel and antiparallel states of injector and
detector. (c) SHE data obtained on a GaAs device at $T=30$~K with the
two Hall contacts in the parallel state for $j_x = \pm 5.7 \times
10^{3}$~A/cm$^{2}$ ($\bullet$ and $\circ$) and antiparallel state
($\relbar$).}
    \label{fig:fig1}
\end{figure}

Typical spin valve and Hanle effect curves (see
Ref.~\onlinecite{Lou:NatPhys:2007} for a discussion) on a lateral spin
valve device are shown in Fig.~\ref{fig:fig1}(b).  These data
establish that the FM contacts are sensitive to the spin polarization
generated by spin injection into the channel as well as its dephasing
by precession in an applied magnetic field.  We also establish that
the Fe contacts, which have an easy axis along [110], show sharp and
reproducible switching behavior as well as nearly perfect remanence. 
The hard axis direction is [1$\overline 1$0], which is parallel to the
the $y$-axis in Fig.~\ref{fig:fig1}(a).  The hard axis saturation
field is $\approx 1.5$~kOe.

Since the contacts are magnetized along [110] ($\hat x$), and the spin
polarization generated by the spin Hall effect is expected to be
oriented along [001] ($\hat z$), a field $B_y$ is applied to precess
the spin accumulation into the [110] direction
\cite{Engel:PRL:2007,Crooker:Science:2005}.  We therefore expect to
observe an increase in the spin Hall voltage at low fields followed by
a suppression due to spin dephasing in large fields.  The signal
should reverse sign when $B_y$ is reversed.  Although the spin
accumulation is small (a few $\mu$V), we will show that it can be
identified by the expected dependence on $B_y$.  In practice, however,
there are three significant background effects that obscure the spin
Hall signal: 1) the ordinary Hall effect due to the applied field, 2)
local Hall effects due to the fringe fields generated by the FM
contacts; and 3) voltages due to the small fraction (0.1\%) of the
channel current that is shunted through the Hall contacts.  The first
two backgrounds can be eliminated fairly easily based on expected
symmetries. For example, reversing the magnetizations of both Hall
contacts from the $+ x$ to $- x$ directions does not change the
ordinary Hall voltage but reverses the sign of the spin Hall voltage. 
By taking the difference of two field sweeps with the Hall contacts in
the two different parallel states, the ordinary Hall effect is thus
removed.  Local Hall effects are due predominantly to the
$x$-components of the contact magnetization.  The corresponding fringe
fields, which are in the $\pm z$-direction are {\it even} with respect
to $B_y$, while the spin Hall voltage is odd in $B_y$.  We can
therefore eliminate local Hall effects by retaining only the
components of the signal that are odd with respect to $B_y$.

Data taken on a GaAs sample with a channel current $j_x = \pm 5.7
\times 10^{3}$~A/cm$^{2}$ at $T=30$~K are shown in
Fig.~\ref{fig:fig1}(c) after removing the first two background
contributions. By construction, these data are odd with respect to
$B_y$, and they show extrema at intermediate fields (approximately 250
Oe) as expected.  Although the exact form of these data will be
discussed below, the magnitude of the Hall voltage at these maxima
corresponds to a spin polarization  $P = (n^\uparrow -
n_\downarrow)/(n^\uparrow + n_\downarrow)\approx 1.3\%$ at the sample
edges, where
\begin{eqnarray}
P = \frac{e \Delta V}{\eta P_{Fe}}\frac{3
m^{*}}{\hbar^{2}(3\pi^{2}n)^{2/3}}.
\label{eq:PfromV}
\end{eqnarray}
In this expression, which follows from the usual relationship between
the spin accumulation and the density of states
\cite{Lou:NatPhys:2007}, $P_{Fe}=0.42$ is the spin polarization of Fe
at the Fermi level and $\eta \approx 0.5$ is the interfacial
transparency.
There are, however, additional features in the field sweeps near 1~kOe
that do not reverse sign when the current is reversed, and hence
cannot be due to a Hall effect.  These result from the current that is
shunted through the Fe contacts (and hence has a component
perpendicular to the plane) in combination with tunneling anisotropic
magnetoresistance (TAMR) at the Schottky contact
\cite{Moser:PRL:2007}.  This final background contribution can be
minimized by subtracting the Hall voltage for the two current
directions, as will be done for all subsequent data shown in this
paper.

We have also performed  the same measurements with the FM contacts on
opposite sides of the channel initialized in either of the
antiparallel states $\uparrow\downarrow$ and $\downarrow\uparrow$. 
The data in this case are shown as the solid line in Fig.
\ref{fig:fig1}(c) after removal of all three backgrounds.  This curve
shows no Hall signal, demonstrating that the spin accumulations at
opposite edges of the sample are opposite in sign.

Data taken at different contact separations for the $x =$~0, 0.03,
0.05 and 0.06 devices at $T=30$~K and $j_{x} = \pm 2.9\times
10^3$~A/cm$^2$ are shown in Fig. \ref{fig:fig2}.  The spin Hall
voltage $\Delta V$  has been converted to spin polarization using
Eq.~\ref{eq:PfromV}. The polarization at contacts further from the
edges of the channel shows a field dependence that is qualitatively
similar to that observed at the edges, but with a smaller magnitude
and narrower width.  The devices with a non-zero In concentration $x$
show a smaller spin Hall signal that decays more rapidly with distance
from the edges of the channel.  No spin signal is observed 10$~\mu$m
from the edge for $x=0.03$ or at 6 and 10$~\mu$m from the edge for $x
= 0.05$ and $0.06$. As confirmed by non-local measurements, these
samples have shorter spin diffusion lengths than the GaAs sample. 
Data similar to those shown in Fig.~\ref{fig:fig2} were taken over a
bias range of $j_{x} = 0$~to~$ \pm 5.7 \times 10^{3}$~A/cm$^{2}$ at $T
= 30$~K and a temperature range of $T = 30$~to~$200$~K at $j_{x} = \pm
5.7 \times 10^{3}$~A/cm$^{2}$.
\begin{figure}
    \includegraphics*[width=8.5cm]{Figure2_Garlid.eps}
    \caption{(color on-line) (a) SHE signal (spin polarization $P$
deduced from the Hall voltage $\Delta V$) ($\bullet$) as a function of
magnetic field for the GaAs sample at $T = 30$~K and channel current
$j_{x} = \pm 2.9 \times 10^{3}$~A/cm$^{2}$.  The solid curves
($\relbar$) show fits to all contact separations with a single set of
parameters.  The data obtained at 6 and 10$~\mu$m from the edges are
multiplied by 5.  (b) Data and fits for the In$_{0.03}$Ga$_{0.97}$As
sample under the same bias conditions.  No spin signal is resolved for
contacts 10$~\mu$m from the edge.  (c) Data and fit for the
In$_{0.05}$Ga$_{0.95}$As sample under the same conditions.  No spin
signal is resolved for contacts 6 and 10$~\mu$m from the edge.  (d)
Data and fit for the In$_{0.06}$Ga$_{0.94}$As sample under the same
conditions.  (e) Magnitude of the SHE signal as a function of distance
from the edge of the channel for all four samples, normalized to
$P_0$.  Solid lines show decay of spin polarization with distance as
determined from fit parameters.}
    \label{fig:fig2}
\end{figure}

We now use these data to determine the magnitude and sign of the spin
Hall conductivity.  The first step in this process is to determine the
transverse spin current $j_{s}\hat y$, which can be related to the
steady-state spin polarization $P_{0}$ at the channel edges by the
diffusion equation, so that $j_s = e P_0 n L_{s}/\tau_{s},$ where
$\tau_s$ is the spin relaxation time and $L_s = \sqrt{D\tau_s}$ is the
spin diffusion length.  Determining $P_0$ requires a full fit of
$P(B_y)$ to a model that includes precession, diffusion, and spin
relaxation.  This is essentially identical to the usual analysis of
non-local Hanle measurements in a lateral spin valve
\cite{Lou:NatPhys:2007} after accounting for the perpendicular
orientation of the ``source'' (the spin Hall current) with respect to
the detector.  To constrain the fits, the diffusion constant $D$ is
obtained from the channel conductivity $\sigma_{xx}$ and the carrier
density $n$ using the Einstein relation \cite{Flatte:PRL:2000}, and
the $g$-factor for each sample is fixed using the value for GaAs, $g =
-0.44$ and the dependence on $x$ determined from the $8 \times 8$ Kane
model
\cite{Silva:PRB:1997,Kiselev:PRB:1998,Madelung:1996,Kato:Science:2004}
.  The modeling also includes the rotation of the magnetization by the
applied field.  This leaves $P_0$ and $\tau_s$ as the only fitting
parameters.  For each bias current, a single set of parameters is used
to fit the data sets obtained at different distances from the edge of
the channel.  The fitting results are shown as solid curves in the
seven panels of Fig.~\ref{fig:fig2}(a-d), and $P$ as a function of
position for all four samples is shown in Fig.~\ref{fig:fig2}(e).  The
principal features of the data are captured by the fitting, including
the decrease with separation from the edges and the location of the
extrema, which shift towards smaller field as the Hall contacts are
moved towards the center of the channel.  Despite the increased
g-factor, the curves for the In$_{x}$Ga$_{1-x}$As samples are broader
than for the GaAs sample.  This reflects the shorter spin relaxation
as $x$ increases, as verified by spin injection measurements.

From the values of $P_{0}$ and $\tau_{s}$ determined from these fits,
it is possible to determine $j_{s}$ and the spin Hall conductivity
$\sigma_{SH} = j_{s}/E_{x}$.  For the GaAs sample, we find
$\sigma_{SH}~\approx~3.0~\Omega^{-1}m^{-1}$, which is of the same
order of magnitude as has been estimated from Kerr microscopy
measurements \cite{Kato:Science:2004,Matsuzaka:PRB:2009} and is of the
same order and sign as has been predicted by theory
\cite{Engel:PRL:2005,Tse:PRL:2006}.  The experimental sign is
determined using the known orientation of the electrode
magnetizations.  To make a more extensive comparison, we consider the
result of Engel and co-workers \cite{Engel:PRL:2005}:
\begin{eqnarray}
\sigma_{SH} \approx
\frac{2\lambda_{so}}{(a^{*}_{B})^{2}}\sigma_{xx}-\frac{2n\lambda_{so}
e^{2}}{\hbar},
\label{eq:SHcondTheory}
\end{eqnarray}
where $\lambda_{so}$ is the spin-orbit coupling parameter, $a^{*}_{B}$
is the effective Bohr radius of an ionized impurity (the presumed
source of scattering), and $\sigma_{xx}$ is the channel conductivity. 
For the case of GaAs, we use $\lambda_{so} = 5.3$~\AA$^2$~and
$a^{*}_{B} = 103$~\AA~(for Si donors) \cite{Engel:PRL:2005}.  At the
value of $j_{x}$ used for the data in Fig.~\ref{fig:fig2},
$\sigma_{xx} = 3600$~$\Omega^{-1}m^{-1}$, and
Eq.~\ref{eq:SHcondTheory} thus gives $\sigma_{SH} =
2.4$~$\Omega^{-1}m^{-1}$, a factor of 20\% smaller than experiment.  A
similar approach was used previously in
Ref.~\onlinecite{Matsuzaka:PRB:2009} to analyze the dependence of
$\sigma_{SH}$ on doping concentration in GaAs. 

As can be seen from examination of Eq.~\ref{eq:SHcondTheory}, there
are two expected contributions to the spin-Hall conductivity, one of
which scales with $\sigma_{xx}$ (skew scattering) and a second which
is a constant of opposite sign (side-jump).  It is possible to tune
the mobility, and hence $\sigma_{xx}$, by approximately 25\% by
varying the bias current \cite{Oliver:PR:1962}.  We find clear
evidence for both contributions in the observed dependence of
$\sigma_{SH}$ on $\sigma_{xx}$, which is shown in
Fig.~\ref{fig:fig3}(a).  The solid lines in this figure are linear
fits.  A negative intercept, indicating the correct sign of the
side-jump term, is found in all four samples.  The slopes and
intercepts of the fits are given in Table~I.  For the purposes of
comparison with theory, we write the total spin Hall conductivity as
$\sigma_{SH} = \sigma_{SS} + \sigma_{SJ} = \gamma\sigma_{xx} +
\sigma_{SJ}$, where $\gamma$ and $\sigma_{SJ}$ are obtained from
Eq.~\ref{eq:SHcondTheory}.   The skewness parameter $\gamma$ for GaAs
is about four times larger than the prediction of $\sim 1 \times
10^{-3}$ \cite{Engel:PRL:2005}.  We can also compare our experimental
results with the expected ratio $\sigma_{SJ}/\gamma = -(a^{*}_{B})^2 n
e^2 \hbar$, which is independent of the spin-orbit coupling
$\lambda_{so}$.  We note that there are large systematic errors
(particularly those originating from the assumption of a Pauli-like
density of states and a fixed $\eta = 0.5$) which appear in the
conversion from a Hall voltage to a polarization (Eq.~\ref{eq:PfromV})
and then to a spin current.  These, however, do not impact the ratio
$\sigma_{SJ}/\gamma,$ which is about a factor of 2.5 larger than the
expected value for all four samples, as shown in Table~I.  
  
\begin{figure}
    \includegraphics*[width=8.5cm]{Figure3_Garlid.eps}
    \caption{(color on-line) (a) Dependence of $\sigma_{SH}$ on
$\sigma_{xx}$ at $T = 30$~K for the $x = $~0 (squares), 0.03
(circles), 0.05 (triangles), and 0.06 (diamonds) samples. A linear fit
is used to determine $\gamma$ and $\sigma_{SJ}$ for each sample. The
fitting parameters are compiled in Table~I.  (b) Temperature
dependence of $\sigma_{SH}$ from $T = 30$~to~$130$~K (solid points)
and predicted temperature dependence (curves).}
    \label{fig:fig3}
\end{figure}

\begin{table}
    \centering
    \caption{Fit parameters for $\sigma_{SH}$ vs. $\sigma_{xx}$.}
        \begin{tabular}{|c|cccc|}
    \hline
$\sigma_{SH} = \gamma\sigma_{xx} + \sigma_{SJ}$  &  ~$0\%$~In~   &  
~$3\%$~In~  & ~$5\%$~In~ & $6\%$~In \\
    \hline
$\gamma$   & 0.004 & 0.003 & 0.012 & 0.011 \\
$\sigma_{SJ}~(\Omega^{-1}m^{-1})$   & -12 & -9 & -35 & -28 \\
Meas. $\sigma_{SJ}/\gamma~(10^3~\Omega^{-1}m^{-1})$   & -3.0 & -3.0 &
-2.9 & -2.2 \\
Pred. $\sigma_{SJ}/\gamma~(10^3~\Omega^{-1}m^{-1})$ & -1.1 & -1.4 &
-1.2 & 0.8 \\
    \hline
    \end{tabular}
\end{table}

We studied the temperature dependence of the SHE in the $x =$~0, 0.03,
and 0.05 samples over the range $T = 30$~to~$150$~K at $j_{x} = 5.7
\times 10^{3}$~A/cm$^{2}$.  Figure \ref{fig:fig3}(b) shows the
experimentally determined spin Hall conductivity as a function of
temperature (solid points).  The solid lines show the predicted
temperature dependence of $\sigma_{SH}$ using the values of $\gamma$
and $\sigma_{SJ}$ determined from Fig. \ref{fig:fig3}(a) and the
measured values of $\sigma_{xx}(T)$ and $n(T)$.  We find that
$\sigma_{SH}$ shows a modest increase over the temperature range of $T
= 30$~to~$100$~K due to the increase in electron mobility.  At
temperatures above $\approx 120$~K the measured spin Hall polarization
decreases rapidly due to the rapid decrease of $\tau_{s}$ with
increasing temperature.  This suppresses the spin Hall signal even if
$\sigma_{SH}$ is relatively constant.  These data suggest that the
spin Hall conductivity is relatively insensitive to phonon scattering,
although it is uncertain whether the extrapolated values above
$\approx 120$~K are accurate.  The weaker temperature dependence of
$\tau_s$ in other materials, such as ZnSe, makes the SHE more readily
observable at high temperatures \cite{Stern:PRL:2006}.

The measurements and analysis presented here conclusively demonstrate
electrical detection of the direct SHE in Fe/In$_{x}$Ga$_{1-x}$As
heterostructures.  The bias and temperature dependences of the SHE
indicate that both skew and side-jump scattering contribute to the
total spin Hall conductivity.  The ratios of the side-jump to skew
scattering contributions for the four samples are similar but larger
than predicted for ionized impurity scattering alone.  Although the
spin Hall conductivity increases with In concentration, this cannot be
attributed unambiguously to an increase in the spin orbit coupling. 
We note, however, that the spin accumulation due to the SHE observed
for the higher In concentrations is comparable to that generated by
direct spin injection from a FM.  This suggests that the SHE could
function as a tool for probing spin-dependent phenomena in materials
with large spin-orbit coupling and short spin diffusion lengths.

We acknowledge helpful discussions with M.~E.~Flatt\' e.  This work
was supported by the ONR MURI program and NSF Grant No. DMR-0804244,
and in part by the NSF MRSEC program under Grant No. DMR-0819885.

%\bibliography{EG_Citation_Database_v7}
\begin{thebibliography}{22}
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\providecommand{\bibinfo}[2]{#2}
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\end{thebibliography}

\end{document}
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